Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-11T14:13:01.421Z Has data issue: false hasContentIssue false
  • English
  • Français

The Asymptotic Posterior Normality of the Latent Trait for Polytomous IRT Models

Published online by Cambridge University Press:  01 January 2025

Hua-Hua Chang
Hua-Hua Chang*
Affiliation:
Educational Testing Service
*
Requests for reprints should be sent to Hua-Hua Chang, Educational Testing Service, Rosedale Road, Princeton, NJ 08541.
Get access Rights & Permissions [Opens in a new window]

Abstract

Chang and Stout (1993) presented a derivation of the asymptotic posterior normality of the latent trait given examinee responses under nonrestrictive nonparametric assumptions for dichotomous IRT models. This paper presents an extention of their results to polytomous IRT models in a fairly straightforward manner. In addition, a global information function is defined, and the relationship between the global information function and the currently used information functions is discussed. An information index that combines both the global and local information is proposed for adaptive testing applications.

Keywords

consistencyefficiencyitem response theorypolytomous itemFisher informationKullback-Leibler informationordered categoriesitem category response function
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 3 , September 1996 , pp. 445 - 463
Copyright
Copyright © 1996 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was partially supported by Educational Testing Service Allocation Project No. 79424. The author wishes to thank Charles Davis, Xuming He, Frank Jenkins, Spence Swinton, William Stout, Ming-Mai Wang, and Zhiliang Ying for their helpful comments and discussions. The author particularly wishes to thank the Editor, Shizuhiko Nishisato, the Associate Editor, and three anonymous reviewers for their thoroughness and thoughtful suggestions.

References

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 516–573.Google Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when the responses are scored in two or more nominal categories. Psychometrika, 37, 29–51.Google Scholar
Chang, H.-H. (1994). Some asymptotic results of polytomous IRT models, Princeton, NJ.: Educational Testing Service.Google Scholar
Chang, H.-H., Stout, W. F. (1991). The asymptotic posterior normality of the latent trait in an IRT model, Urbana-Champaign: University of Illinois, Department of Statistics.CrossRefGoogle Scholar
Chang, H.-H., Stout, W. F. (1993). The asymptotic posterior normality of the latent trait in an IRT model. Psychometrika, 58, 37–52.Google Scholar
Chang, H.-H., & Ying, Z. (in press). A global information approach to computerized adaptive testing. Applied Psychological Measurement.Google Scholar
Cover, T. M., Thomas, J. A. (1991). Elements of information theory, New York: John Wiley & Sons.Google Scholar
Holland, P. W. (1990). On the sampling theory foundations of item response theory models. Psychometrika, 55, 577–601.Google Scholar
Johnson, E. G., Carlson, J. E. (1994). The NAEP 1992 technical report, Washington, DC: National Center of Education Statistics.Google Scholar
Junker, B. W. (1991). Essential independence and likelihood based ability estimation for polytomous items. Psychometrika, 56, 255–278.Google Scholar
Lehmann, E. L. (1983). Theory of point estimation, New York: John Wiley & Sons.Google Scholar
Lord, M. F. (1971). Robbins-Monro procedures for tailored testing. Educational and Psychological Measurement, 31, 3–31.Google Scholar
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.Google Scholar
Muraki, E. (1992). A generalized partial credit model: application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.Google Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement No. 17, 34 (4, Pt. 2).Google Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph No. 18, 37 (4, Pt. 2).Google Scholar
Serfling, R. J. (1980). Approximation theorems in mathematical statistics, New York: John-Wiley & Sons.Google Scholar
Thissen, D., Steinberg, L. (1984). A response model for multiple choice items. Psychometrika, 49, 501–519.Google Scholar
Thissen, D., Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577.Google Scholar
Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Annals of Mathematical Statistics, 20, 595–601.Google Scholar
Walker, A. M. (1969). On the asymptotic behavior of posterior distributions. Journal of the Royal Statistical Society, Series B, 31, 80–88.Google Scholar
Weiss, D. J. (1982). Improving measurement equality and efficiency with adaptive testing. Applied Psychological Measurement, 6, 379–396.Google Scholar
Wilson, M., Masters, G. N. (1993). The partial credit model and null categories. Psychometrika, 58, 87–99.Google Scholar
Wright, B., Masters, G. N. (1982). Rating scale analysis, Chicago: MESA Press.Google Scholar